Hi there. Well, I have some doubts about this exercise, I think I've solved it, but I wanted your opinion, which always help. So, it says:

There is no general method for solving the homogeneous general equation of second order

$\displaystyle y''+P(x)y'+Q(x)y=0$ (1)

But if we already know a solution $\displaystyle y_1(x)\neq 0$, then we always can find a second solution linearly independent

$\displaystyle y_2(x)=y_1(x) \int \dysplaystyle\frac{e^{-\int P(x)dx}}{y_1^2}dx$

Demonstrate that this functions form a basis for the space of solutions of the differential equation (1).

So, what I did is simple. I know that if the solution is linearly independent, then the Wronskian $\displaystyle w(y_1,y_2,x)$ must be zero. So, I just made the calculus for the wronskian:

$\displaystyle \left| \begin{matrix}{y_1}&{y_2}\\{y_1'}&{y_2'}\end{matri x} \right|=\left| \begin{matrix}{y_1}&{y_1(x) \int \dysplaystyle\frac{e^{-\int P(x)dx}}{y_1^2}dx}\\{y_1'}&{y_1'(x) \int \displaystyle\frac{e^{- \int P(x)dx}}{y_1^2}dx+y_1(x) \displaystyle\frac{e^{- \int P(x)dx}}{y_1^2(x)}dx}\end{matrix} \right|=\displaystyle\frac{e^{- \int P(x)dx}}{y_1(x)}\neq0$

Is this right?

Bye there.